Berezinskii-Kosterlitz-Thouless phases in ultrathin ${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ superlattices

We study the emergence of Berezinskii-Kosterlitz-Thouless (BKT) phases in ${\left({\mathrm{PbTiO}}_3\right)}_3/{\left({\mathrm{SrTiO}}_3\right)}_3$ superlattices by means of second-principles simulations. Between a tensile epitaxial strain of $\epsilon =0.25–1\%$, the local dipole moments within the superlattices are confined to the film-plane, and thus the polarization can be effectively considered as two-dimensional. The analysis of the decay of the dipole-dipole correlation with the distance, together with the study of the density of defects and its distribution as a function of temperature, supports the existence of a BKT phase in a range of temperature mediating the ordered ferroelectric (stable at low $T$), and the disordered paraelectric phase that appears beyond a critical temperature $T_{\mathrm{BKT}}$. This BKT phase is characterized by quasi-long-range order (whose signature is a power-law decay of the correlations with the distance), and the emergence of tightly bound vortex-antivortex pairs whose density is determined by a thermal activation process. The proposed ${\mathrm{PbTiO}}_3/{\mathrm{SrTiO}}_3$ superlattice model and the imposed mechanical boundary conditions are both experimentally feasible, making it susceptible for an experimental observation of these new topological phases in ferroelectric materials.

Figure 1

Polarization distribution as a function of the epitaxial strain. Distribution of the local dipole components at 25 K under a tensile epitaxial strain of (a) 0.25%, (b) 1.00%, and (c) 3.00%. Red, blue, and green histograms represent, respectively, the local dipole components along [100], [010], and [001] pseudocubic directions. The magnitude of the dipole moment $d$ is given in atomic units. Panels (d)–(f) represent the electric polarization textures, following the tensile epitaxial strain of the top row panels. Black arrows indicate the local dipoles, projected onto the $(x,y)$ plane. The out-of-plane component of the polarization along the ${\left[001\right]}_{\mathrm{pc}}$ direction is represented by the green and magenta domains.

Figure 2

Two different representations of the correlation function as a function of the distance for a supercell of size $L=20$ under $\epsilon =0.25\%$ tensile epitaxial strain. (a) Logarithm of the correlation function against distance. (b) Logarithm of the correlation function against the logarithm of the distance. Symbols for different temperatures are shown in the inbox. u.c. stands for unit cell.

Figure 3

Correlation functions vs the dipole-dipole distance for different temperatures and sizes of the simulation box at constant epitaxial strain of $\epsilon =3\%$. (a) $C\left(r\right)$ as a function of $r$ for $T=430\mathrm{K}$. (b) Same data, but in log-log scale. The same black solid line in panels (a) and (b) represents a linear fit of the data obtained for the largest simulation box. (c) $C\left(r\right)$ as a function of $r$ for $T=480\mathrm{K}$. (d) Same data, but taking the logarithm of the correlation function. The same solid black line in panels (c) and (d) represents a linear fit for the data obtained at $L=30$ below a cutoff radii of 4.5 unit cells. Different lattice sizes $L=10,20,30,40$ are represented by red squares, blue squares, green triangles, and orange diamonds, respectively. u.c. stands for unit cell.

Figure 4

Two different representations of the dipole-dipole correlation as a function of the distance for a supercell of size $L=20$ under different mechanical boundary conditions. ($\mathrm{a})\epsilon =1\%$. Logarithm of the correlation function against the distance. (b) $\epsilon =1\%$. Logarithm of the correlation function against the logarithm of the distance. (c) $\epsilon =3\%$. Logarithm of the correlation function against the distance. (d) $\epsilon =3\%$. Logarithm of the correlation function against the logarithm of the distance. Symbols for different temperatures are shown in the inbox. The dashed-oval at the right-hand side encapsulates the temperature range where the results from the simulations can be fitted to a straight line, with a regression coefficient larger than 0.97.

Figure 5

Distribution of topological defects for a supercell of size $L=20$ and a tensile epitaxial strain of $\epsilon =1\%$ at different temperatures. (a) $T=220\mathrm{K}$, (b) $T\approx T_{\mathrm{BKT}}=255\mathrm{K}$, and (c) $T=290\mathrm{K}$. Red dots represent vortices, with vorticity $n=+1$, whereas blue stars stand for antivortices of vorticity $n=-1$.

Figure 6

Density of vortex-antivortex pair defects as a function of temperature for a supercell of size $L=20$. Red dots and blue squares correspond to a tensile epitaxial strain of $\epsilon =1\%$ and $3\%$, respectively. Vertical dashed line is a guide to the eye for identifying the transition temperature at the point where the curvature changes. Black solid lines are a fit to a linear regression where the two last points are omitted in the case of $\epsilon =1\%$ as they correspond to 180 and 190 K, where an ordered state is observed as shown in the steady plateau of Fig. 4.

Figure 7

Sketch of a possible configuration of dipoles with vorticity $n=+1$ along an elementary plaquette of the simulation cell centered at the site $(i,j)$. The direction of the red arrow marks the sense of travel in order to compute the vorticity.